gjm

Mathematician. Reader. Fiftysomething. Cruciverbalist. Not very good at thinking of witty things to put in short bios.

I wasn't on Twitter before that nice Mr Musk suggested that everyone go to Mastodon, and I don't expect to be very active here, but who knows?

Public Key

npub19ym8s0u4ahcxwtu0qrhpnp0rnlamhem33qae82mmsgt47pjv05aqp8lqjq

NIP-05 Address

gjm@mathstodon-xyz.mostr.pub

Profile Code

nprofile1qqszjdnc8727mur8978spmsesh3el7amuaccswun4dacy96lqex86wspz3mhxue69uhhyetvv9ujuerpd46hxtnfduqs6amnwvaz7tmwdaejumr0dssf2djz

Publishing to

relay.damus.io

nos.lol

Author Public Key

npub19ym8s0u4ahcxwtu0qrhpnp0rnlamhem33qae82mmsgt47pjv05aqp8lqjq

Show more details

Last Notes

2026-07-23 15:36:11 UTC

- reply

@nprofile…tkx6 @nprofile…eljp I'm not pozorvlak but I don't think there's any call to delete the quoot. The basic point is: the AI labs have models that (1) are really quite capable and (2) are demonstrably susceptible to paperclip-maximization-type behaviour, and we should be alarmed by this.
We should maybe be a little less alarmed by it if the paperclippiest models are only running inside the AI labs, but given that we just saw one of them getting out and having consequences outside the AI lab in question I don't personally feel terribly reassured by that.

2026-04-17 10:29:32 UTC

- reply

@nprofile…tkx6 On the other hand, the image itself is also largely unnecessary given the text it's accompanying; all it does is to give a bit of evidence that the statement in the text is true and to show, for those able to see it, what it actually looks like.
Actually-useful alt-text for this image would pretty much just duplicate the information that's already there in the original text.
(I think there is such a thing as an image that can't usefully be alt-texted because its _only_ purpose is to provide a visual impression of something that's already been explained textually as well as the person writing the text knows how to do. Such images should in fact still have alt-text, to let visually impaired readers know what sort of image it is. "Image of a female character from The Elder Scrolls, from near ground level, showing the smiley face described in the text" or something.)
I suspect that the LLM-based auto-alt-text generators could in fact do this sort of thing pretty well if given the accompanying text as well as the image, and a suitable prompt.
[EDITED to add:] I had a look at altbot, which I think is the thing that does this. Its prompt specifically tells the LLM just to describe what it sees without interpreting -- which may well be the _correct_ thing to do when there is actual information in the image not already present in the text, even though in cases like this where there really isn't the result is something that wears its uselessness on its sleeve.
(Actually, _is_ it completely useless? If I had read the text and couldn't see the image, it feels like this auto-alt-text might tell me about as much about the image as could usefully be conveyed in text.)

2026-03-10 13:38:06 UTC

- reply

@nprofile…tkx6 I wonder how much alcohol bought in shops is for _solo_ drinking as opposed to _drinking with one's family_ and how Anon Opin feels about the latter relative to "social drinking" and "private drinking", which it seems kinda intermediate between.
I also wonder whether social drinking is a social good, or whether _socializing_ is a social good and drinking is just a thing that tends to accompany it. If alcohol in restaurants were cheaper, people would probably eat out about the same amount, although they might go to marginally fancier restaurants or something. Cheaper alcohol in pubs probably _would_ mean more people going for a drink in a pub, but would it mean more socializing or just make some people switch from other social activities?
(It could well be that social drinking as such _is_ beneficial -- being a bit more relaxed and disinhibited probably helps with making and building friendships, and maybe it helps more on balance than having slightly more drunken arguments and fights hurts.)

2026-02-10 22:38:19 UTC

- reply

@nprofile…4hp5 Are you sure about your Juliuses and your Scaligers?
It looks to me as if:
Julius Caesar (the emperor) introduced the Julian calendar.
Julius Caesar Scaliger was an erudite chap but had nothing much to do with calendars.
Joseph Justus Scaliger, Julius Scaliger's son, introduced the "Julian period" and (kinda) the "Julian day number".
The "Julian" there is a reference to the Julian calendar, _not_ to his dad.
... But I am not an expert, I'm just reporting what I find in Wikipedia, and maybe the above is all wrong?

2026-01-24 20:28:23 UTC

- reply

@nprofile…g0zc @nprofile…4hp5 If you mean "feck" as in the thing that "feckless" means not having, then of course they are cognate. But if you mean "feck" as in "minced version of another word differing only in its vowel" then that doesn't seem to be true; that other word has an origin entirely different from that of "feck(less)".
(And unless I've misunderstood very badly, the context was all about feck-the-minced-oath versus feck-as-in-less.)

2026-01-19 16:08:11 UTC

- reply

@nprofile…g0zc You can use +n or -n and have it still work, no? Just offset the value of n by 1.

2026-01-19 16:03:49 UTC

- reply

@nprofile…g0zc That's very clever!
(It's Euler's 𝑛²+𝑛+41 thing in disguise. And of course it just happens to break down immediately after that 01601 block.)

2026-01-16 12:26:30 UTC

- reply

@nprofile…tfwq I agree that you probably want different scoring methods for different contexts. My first thought about a generic approach: suppose you have 𝑘 objects and can have each one up to 𝑚ₖ times; think of that as actually having ∑𝑚ₖ objects, and always picking the copies of each in the same order. Now the person being assessed is making ∑𝑚ₖ binary decisions and you can give them one mark for each one they make correctly. (Special case: if repetitions aren't allowed, you're giving them one point for each object they classify correctly. Which seems reasonable.)
But, again, as you say there will be situations where this is entirely the wrong thing; e.g., maybe some objects are good substitutes for others, maybe it's extra-bad to pick too many, maybe missing them is catastrophic but picking extra ones much less so, etc.

2026-01-15 17:31:01 UTC

- reply

@nprofile…tkx6 Is it possible that this one is actually not expressing a sincere opinion but making fun of people who want to ban things for no adequate reason?
(Alas, I suspect probably not.)

2026-01-14 12:34:10 UTC

- reply

@nprofile…g0zc Huh, how come the car's red lights are in the shape of the word OBEY?

2025-12-27 16:01:20 UTC

- reply

@nprofile…um96 It looks like you've got two copies of the script appended to one another?

2025-12-18 18:41:11 UTC

- reply

@nprofile…l4xy Schramm was a damn good mathematician. It's very sad that he died so young. (I guess it would be very sad that he died so young even if he'd been a rotten mathematician, but you know what I mean.)

2025-12-10 23:40:00 UTC

- reply

@nprofile…vf9x I did a bit of searching and found a book from 2002 ("Dinosaurs of the Air", by Gregory Paul) that seems to be reasonably scholarly and that (1) says -- hence, in particular, _finds it necessary to say_ -- that the idea that dinosaurs are birds is supported by an overwhelming preponderance of evidence, (2) quotes a slightly earlier book (from 1996) that argues that birds _aren't_ dinosaurs and calls the idea that they are a "dogma", and (3) contains the following sentence: "Had the complete skulls and articulated skeletons of [four groups of dinosaurs] known today been available earlier in the 1990s, the dinosaur-bird link would long ago have become the established dogma in the paleontological and ornithological communities."
So it seems like the question was genuinely open in the early 1990s, and (at least according to this guy, but so far as I can tell he's describing a basically-consensus position) pretty much settled some time in the early 2000s.
So I think the "one day not too long ago" was somewhere in the late 1990s.
Ooh, found another data point: an article by Thulborn in 1985 called "Birds as neotenous dinosaurs" that says "Recently identified osteological characters in theropods challenge the conventional distinction between birds and dinosaurs."
So: in the early 80s the conventional wisdom was that birds _aren't_ dinosaurs; in the early 90s it was at least somewhat in doubt; by the late 90s it was "dogma" but some questioned it; by the early 00s there wasn't much doubt left.
(I did also find e.g. a popular-science book from 1982 that says "birds are dinosaurs" so it's not like the idea wasn't around earlier. But I don't think it was the usual view then.)

2025-12-04 17:23:04 UTC

- reply

@nprofile…j0re My sleep-deprived brain initially read your last bullet point as "How can I remove all Mayan artifacts from my machine?". Well, first you need a *really sharp* obsidian knife...

2025-11-19 20:01:01 UTC

- reply

@nprofile…l4xy Dunno whether you count this, which is conspicuously not on the list despite being in some sense very hexagonal. (It's Trevelyan College of the University of Durham.)
https://media.mathstodon.xyz/media_attachments/files/115/578/135/917/994/709/original/a4df8640dd85f95d.png

2025-10-09 14:47:54 UTC

- reply

@nprofile…yu7x By "fairly small" I meant on the order of 10^-10, which is the sort of thing I would expect to be fairly easily achievable in multiple ways by things looking like linear combinations of spherical shells.

2025-10-09 12:49:30 UTC

- reply

@nprofile…yu7x Note that the gravitational field of _any_ spherically-symmetrical thing is the same (up to a scaling of the total mass) _everywhere_ outside the sphere containing it. So it doesn't particularly matter whether the sample points are all on a single sphere.
Of course, that's what's true for an _actually literally spherically symmetrical_ mass density, and what you've got here is just an approximation to that. I don't think I'd have predicted ahead of time that the determinant would be so very small ... though maybe what's going on isn't that there are _incredibly small eigenvectors_ but that there are _quite a lot of fairly small eigenvectors_.
Further conjecture: if you move your sample points further out, the determinant will get smaller; if you move them further in, it will get larger. (On the grounds that I expect the spherically-symmetrical approximation to get better at larger distances.)

2025-10-09 12:22:50 UTC

- reply

@nprofile…yu7x Have you looked to see what the almost-nullspace looks like? That is, what not-very-short vectors are mapped almost-to-zero? Or, to put it differently: your empirical finding shows that some distributions of mass have almost the same effect on your 83 red dots as others; what are they?
My best guess, which I don't trust much: your 83 red dots are all at a single distance, so any distribution of mass that looks "enough" like a spherical shell of any size smaller than that will look (up to scaling the masses) exactly like any other, and perhaps your nearly-nullspace vectors look like linear combinations of spherical shells.

2025-06-16 01:04:39 UTC

- reply

@nprofile…saev I have seen "directly congruent" (and, likewise, "directly similar"). It's fairly easy to guess what those mean if you don't happen to know.
I really don't like "conguent", which will leave readers who know the distinction you're making unsure whether you're making it or have just made a typo, and readers who _don't_ just thinking you made a typo.
(I'm not a big fan of "rngs" and "rigs" and the like, either.)

2025-06-15 13:21:05 UTC

- reply

@nprofile…fa4k @nprofile…mqrg @nprofile…6cfc @nprofile…hszg @nprofile…mlph That paper also contains the line "the d-ism of Leibniz overtook the dotage of Newton" which is rather magnificent. Knuth puts it in quotes but doesn't say who it's from; it seems to be Charles Babbage, though his wording was a little different.

2025-06-12 20:57:06 UTC

- reply

@nprofile…saev @nprofile…e8p2 @nprofile…6al5 Every day is @pigworker appreciation day.

2025-06-11 20:02:13 UTC

- reply

@nprofile…hr6z Aperiodic _tilings_ but not aperiodic _tiles_. That is, their tiles can tile the plane periodically as well as aperiodically.
My understanding is that what's claimed to be new is something about the particular aperiodic patterns these tiles can tile the plane in.

2025-05-29 13:37:21 UTC

- reply

@nprofile…hr6z Looks like you're writing it in COBOL.
TOKEN SECTION.
CALL 'INITIALIZE-PARSER'.
ADD ONE TO CHARACTER-COUNT.
...
END OF TOKEN SECTION.

2025-05-14 22:10:17 UTC

- reply

@nprofile…saev I think they're trying to clarify that they can do 4x4 complex matmul with 48 complex multiplies but _not_ 4x4 real matmul with 48 real multiplies, which is a reasonable thing to clarify. But while they were clarifying that they should also have clarified whether their method works with (complex) matrix coeffs and hence can be used recursively.

2025-05-14 21:07:22 UTC

- reply

@nprofile…saev Further, a comment on the MSE answer (whose correctness I have made no attempt to check) says that the 48-scalar-multiplication 4x4 matmul described there _assumes the matrix entries are in a commutative ring_ which means you can't use it recursively to get a better exponent than Strassen, which (assuming you can for the Deep Mind one, which I also haven't checked) seems like a very important distinction.

2025-04-30 20:36:08 UTC

- reply

@nprofile…59k6 Well, that's why I gave groups as an example -- the notion doesn't really need any prior knowledge, and you ought to be able to get sufficient motivation from the idea by thinking about geometry, which the ancient Greeks were very keen on.
(Though historically group theory came out of thinking about roots of polynomials, not so much an ancient Greek thing although there's a certain amount in common between "constructible with straightedge and compass" and "solvable by radicals".)

2025-04-30 15:16:48 UTC

- reply

@nprofile…59k6 There are similar things in pure mathematics, which by definition isn't technology-limited (although some things in pure mathematics are _motivated_ by empirically-influenced scientific advances, and some are _impractical_ without high-speed automated calculating devices). For instance, there's no particular reason why the ancient Greeks shouldn't have discovered/invented group theory, but they didn't.
Obviously "group theory" and "role-playing games" are quite different kinds of thing! But they _are_ both examples of good ideas that could have arisen much earlier than they did.

2025-02-22 00:24:48 UTC

- reply

@nprofile…cas5 @nprofile…fqqu @nprofile…q8c7 There are plenty of solutions to just 𝑎²+𝑏²=…=𝑀² -- the tricky thing is getting those extra two conditions to hold as well.
(Solving the equations without the extra two conditions basically comes down to finding numbers with "enough" factors in the so-called Gaussian integers, things of the form a + ib where i^2 = -1. Any time you have (𝑎+𝑖𝑏)(𝑎−𝑖𝑏)=𝑀² you also have 𝑎²+𝑏²=𝑀² . It's possible to get many many such factors, i.e., for that circle to have lots of rational points on it. But it's more difficult when we have the extra constraints relating those different (a,b) pairs.)

2025-02-08 13:50:57 UTC

- reply

@nprofile…59k6 @nprofile…yu7x I think the characteristic of Gödel that's relevant to the story is less "genius mathematical logician" and more "person inclined to treat things as puzzles in mathematical logic".

2025-01-30 15:12:58 UTC

- reply

@nprofile…hujc @nprofile…t7j4 @nprofile…5sws If you suspect that an author has plagiarized another, there are ways of getting concrete evidence: look at the other work and the new work, and compare. If you suspect they've plagiarized but don't know what, you may be able to find their victim by searching.
But if you suspect that something was generated with AI, it's not so easy to check. It's as if you suspect that someone has plagiarized ... but from a huge corpus of work to which you have no access.
On the other hand: if you genuinely can't tell, maybe it's time to stop worrying. That is: we prefer human-generated works to AI works because they're better written, or more accurate, or less plagiarized; if a book is badly written or wrong or plagiarized then we can discern _that_ and write scathing reviews, not buy it, etc. And the way we catch fake "human generated" certs is by noticing that a book has that cert but has the flaws that an AI-written book would have.
But if at some point AIs are able to write books that don't have those detectable problems ... well, then, maybe at that point there isn't much need for a "human generated" cert any more.

2025-01-17 21:26:08 UTC

- reply

@nprofile…hujc The explanation that comes to mind is that CSL decided that he wanted those changes, and a new US edition was an opportunity to make them. In that case you'd expect subsequent UK editions to have the same changes, though, and I'm guessing that that isn't the case.
Maybe at the time of publication some other Norse-flavoured thing was popular in the US, and the US editors said "can we Norse it up a bit?" and CSL said "sure, how about this?"?
Everything above is baseless guesswork, of course. But it seems likely that there's something a bit more than "US editors gratuitously made a few random changes" going on.

2025-01-08 00:22:01 UTC

- reply

@nprofile…hujc I like almost everything in that piece very much (well, maybe "like" isn't the right word -- I'm not wild about the state of affairs he's describing -- but I think he's saying true and insightful things). But one thing strikes me as odd. Within a couple of paragraphs, he is indignant (1) that Aaron Swartz was hounded by the law for "downloading too many articles from JSTOR" and (2) that Sam Altman's company is scraping the entire internet and thinks that "copyright law is only for little people".
Like, if the problem with what OpenAI is doing is that it pushes the boundary of the law around how computers are allowed to access information, then it's hard not to notice that that's also what Swartz got jumped on for.
There are differences, and they're important ones -- OpenAI is pushing those boundaries for its own profit (I don't find myself very convinced by their rhetoric about trying to make AGI benefit humanity) while Swartz was doing it because he believed research papers should be free for everyone to read (I think?), and there are lots of ways in which Swartz seems to have been an excellent person and Altman, well, not so excellent. But I think Karpf's rhetoric here is inconsistent and self-undermining.
(I take the view that training AI systems on copyrighted materials is no worse than training humans on copyrighted materials. The problem, if there is one, arises when they start reproducing stuff they're not allowed to, not when they learn it. Same as for humans.)

2024-12-19 22:09:25 UTC

- reply

@nprofile…9u60 I _think_ this isn't a website being scammy about cookies, but an outright malware-installation-attempt attack. If you click "allow" you're getting something worse than cookies. (See e.g. https://www.reddit.com/r/Scams/comments/17vh4c8/went_to_go_download_something_off_of_y2mate_this/.)

2024-12-06 02:23:33 UTC

- reply

@nprofile…saev _Biggest possible_ if true.

2024-12-02 21:18:59 UTC

- reply

Whether or not _that_ was a joke, there's something very appropriate about your photo showing Aristote's Fragme Sele.

2024-08-29 17:03:00 UTC

- reply

@npub1c9m…9xjd Possibly almost as good as a spoon of honey without turmeric sprinkled on top.

2024-07-17 18:20:16 UTC

- reply

@npub1c9m…9xjd I feel that they should have found some suitable definition that would let them label the figure "The _wiggly_ bijection between wiggly pseudotriangulations and wiggly permutations".

2024-07-05 11:03:04 UTC

- reply

@npub1gpa…xlk0 Final outcome seems to be better (at least according to your preferences and also mine) than the original prediction: the Conservatives have more like 120 seats, the Lib Dems more like 70, and Reform still only 4.

2024-06-18 19:56:21 UTC

- reply

@npub1psd…9sr7 https://www.oglaf.com/blank-page/
(warning: is Oglaf. This one is fairly SFW by Oglaf's standards.)

2024-05-06 00:43:11 UTC

- reply

@npub1c9m…9xjd The only time I've had an MRI they never asked me anything of that sort.